Solution Set For Inequality 5(x-2)(x+4) > 0

Navigating the world of inequalities can sometimes feel like traversing a complex maze. But fear not, for with the right tools and techniques, we can systematically unravel even the most intricate problems. Today, we embark on a journey to decipher the solution set for the inequality 5(x-2)(x+4) > 0, a quintessential problem in algebra that tests our understanding of inequalities and quadratic expressions. This exploration will not only equip you with the ability to solve this specific problem but also provide a framework for tackling similar challenges in the future. Let's dive in and demystify the process, ensuring you grasp every step along the way.

Understanding the Inequality: A Foundation for Success

Before we jump into the mechanics of solving the inequality 5(x-2)(x+4) > 0, it's crucial to lay a solid foundation by understanding the core concepts at play. At its heart, an inequality is a mathematical statement that compares two expressions, indicating that they are not necessarily equal. In this case, we're dealing with a quadratic inequality, where the expression involves a polynomial of degree two. The significance of the '> 0' part of the inequality is that we are searching for the values of 'x' that make the expression on the left-hand side strictly positive. This means that we're not interested in values of 'x' that make the expression equal to zero; we only want the ones that yield a positive result. The expression 5(x-2)(x+4) is a product of three factors: the constant 5, and the linear expressions (x-2) and (x+4). Understanding the behavior of these factors is key to solving the inequality. Each factor contributes to the overall sign of the expression, and it's the interplay of these signs that ultimately determines whether the inequality holds true. By focusing on the individual factors and how they change sign, we can systematically identify the regions on the number line where the inequality is satisfied. This approach not only simplifies the problem but also provides a deeper insight into the nature of inequalities and their solutions.

Finding the Critical Points: The Cornerstones of the Solution

The first crucial step in solving the inequality 5(x-2)(x+4) > 0 involves identifying the critical points. These points are the values of 'x' that make the expression on the left-hand side equal to zero. In essence, they are the boundaries that divide the number line into intervals, each of which we will analyze to determine whether the inequality holds true within that interval. To find these critical points, we set each factor of the expression equal to zero and solve for 'x'. Starting with the factor (x-2), we have x-2 = 0, which yields x = 2. This means that when x is equal to 2, the factor (x-2) becomes zero, and consequently, the entire expression 5(x-2)(x+4) becomes zero. Similarly, for the factor (x+4), we have x+4 = 0, which gives us x = -4. When x is equal to -4, the factor (x+4) becomes zero, again making the entire expression zero. The constant factor 5, being a non-zero constant, does not contribute to the critical points. Therefore, our critical points are x = 2 and x = -4. These two points are the linchpins of our solution strategy. They divide the number line into three distinct intervals: x < -4, -4 < x < 2, and x > 2. In the next step, we will examine each of these intervals to determine the sign of the expression 5(x-2)(x+4) within each interval. This will allow us to pinpoint the intervals where the inequality 5(x-2)(x+4) > 0 holds true.

Interval Analysis: Mapping the Solution Landscape

With the critical points identified as x = -4 and x = 2, the next step in solving the inequality 5(x-2)(x+4) > 0 is to perform an interval analysis. This technique involves examining the sign of the expression 5(x-2)(x+4) within each of the intervals created by the critical points. These intervals are: (-∞, -4), (-4, 2), and (2, ∞). For each interval, we select a test value – a number that falls within the interval – and substitute it into the expression 5(x-2)(x+4). The sign of the resulting value will tell us whether the expression is positive or negative within that interval. Let's begin with the interval (-∞, -4). We can choose a test value such as x = -5. Substituting this into the expression, we get 5(-5-2)(-5+4) = 5(-7)(-1) = 35, which is positive. Therefore, the expression 5(x-2)(x+4) is positive in the interval (-∞, -4). Next, we consider the interval (-4, 2). A suitable test value would be x = 0. Substituting this into the expression, we get 5(0-2)(0+4) = 5(-2)(4) = -40, which is negative. Thus, the expression is negative in the interval (-4, 2). Finally, we examine the interval (2, ∞). A test value like x = 3 works well. Substituting this, we get 5(3-2)(3+4) = 5(1)(7) = 35, which is positive. So, the expression is positive in the interval (2, ∞). By performing this interval analysis, we have effectively mapped the sign of the expression across the number line. We know that 5(x-2)(x+4) is positive in the intervals (-∞, -4) and (2, ∞), and negative in the interval (-4, 2). This information is crucial for determining the solution set to the inequality.

Constructing the Solution Set: Putting the Pieces Together

Having meticulously analyzed the intervals and determined the sign of the expression 5(x-2)(x+4) in each, we are now poised to construct the solution set for the inequality 5(x-2)(x+4) > 0. Recall that we are seeking the values of 'x' that make the expression strictly greater than zero, meaning we are only interested in the intervals where the expression is positive. From our interval analysis, we found that the expression is positive in the intervals (-∞, -4) and (2, ∞). This means that any value of 'x' within these intervals will satisfy the inequality. However, it's crucial to remember that the inequality is strictly greater than zero, not greater than or equal to zero. This subtle distinction has a significant impact on how we represent the solution set. Since the expression equals zero at the critical points x = -4 and x = 2, we must exclude these points from the solution set. If we were dealing with an inequality that included the 'equal to' case (i.e., ≥ 0), we would include these points. But in this case, we use parentheses in our interval notation to indicate that the endpoints are not included. Therefore, the solution set consists of all 'x' values less than -4 or greater than 2. Mathematically, we express this as the union of two intervals: (-∞, -4) ∪ (2, ∞). This notation succinctly captures all the values of 'x' that satisfy the original inequality. It's a precise and unambiguous way to communicate the complete set of solutions.

Expressing the Solution Set: Different Notations, Same Meaning

While we've successfully constructed the solution set for the inequality 5(x-2)(x+4) > 0 using interval notation, it's important to recognize that there are often multiple ways to express the same solution set. Understanding these different notations is crucial for effective communication in mathematics and for interpreting solutions presented in various contexts. We've already established that the solution set in interval notation is (-∞, -4) ∪ (2, ∞). This notation is concise and visually represents the intervals on the number line where the inequality holds true. However, an alternative way to express this solution set is using set-builder notation. Set-builder notation uses a more symbolic approach, defining the solution set as a collection of 'x' values that satisfy a specific condition. In this case, the condition is that 'x' must be less than -4 or greater than 2. Symbolically, we write this as {x | x < -4 or x > 2}. This notation reads as